Express 0.78 as a fraction a/b where a,b ∈ Z Expand and Simplify: (7y2 – 6y – 8)(-2y + 2) Factorise the following expressions fully. 3.1 8x2 - 6x + 1 3.2 125x3 + 1

Okay, dineo❤️! Let's tackle these math problems. 1.1 Express 0.78 as a fraction (a)/(b) where a, b Z Step 1: Let x be the repeating decimal. x = 0.78 Step 2: Multiply by powers of 10 to isolate the repeating part. 10x = 7.8 (1) 100x = 78.8 (2) Step 3: Subtract equation (1) from equation (2). 100x - 10x = 78.8 - 7.8 90x = 71 Step 4: Solve for x and simplify the fraction. x = (71)/(90) The fraction is (71)/(90). 1.2 Expand and Simplify: (7y^2 - 6y - 8)(-2y + 2) Step 1: Distribute each term from the first polynomial to each term in the second polynomial. = 7y^2(-2y) + 7y^2(2) - 6y(-2y) - 6y(2) - 8(-2y) - 8(2) Step 2: Perform the multiplications. = -14y^3 + 14y^2 + 12y^2 - 12y + 16y - 16 Step 3: Combine like terms. = -14y^3 + (14y^2 + 12y^2) + (-12y + 16y) - 16 = -14y^3 + 26y^2 + 4y - 16 The expanded and simplified expression is -14y^3 + 26y^2 + 4y - 16. 1.3 Factorise the following expressions fully. 1.3.1 8x^2 - 6x + 1 Step 1: Find two numbers that multiply to 8 × 1 = 8 and add up to -6. These numbers are -2 and -4. Step 2: Rewrite the middle term using these two numbers. 8x^2 - 2x - 4x + 1 Step 3: Factor by grouping. 2x(4x - 1) - 1(4x - 1) Step 4: Factor out the common binomial factor (4x - 1). (2x - 1)(4x - 1) The factorised expression is (2x - 1)(4x - 1). 1.3.2 125x^3 + 1 Step 1: Recognize this as a sum of cubes, which follows the formula a^3 + b^3 = (a+b)(a^2 - ab + b^2). Step 2: Identify a and b. a^3 = 125x^3 a = 5x b^3 = 1 b = 1 Step 3: Substitute a and b into the sum of cubes formula. (5x + 1)((5x)^2 - (5x)(1) + (1)^2) Step 4: Simplify the terms. (5x + 1)(25x^2 - 5x + 1) The factorised expression is (5x + 1)(25x^2 - 5x + 1). 3 done, 2 left today. You're making progress.